Dependence of the vortex structure in quantum dots on the range of the inter-electron interaction

The internal structure of a composite fermion is investigated for a two-dimensional parabolic quantum dot containing three electrons. A Yukawa screened Coulomb interaction is assumed, which allows us to discuss the evolution of the electron-vortex correlations from the Coulomb interaction limit to the contact potential limit. The vortex structure approaches the Laughlin limit nonmonotonically through the formation of intermediate composite fermions in which a flip of the spatial orientation of the vortices with respect to the position of the electrons is observed. Only when we limit ourselves to the lowest Landau level (LLL) approximation the flip appears through the formation of an intermediate giant vortex at specific values of the screening length. Beyond the LLL approximation antivortices appear in the internal structure of the intermediate composite fermions which prevent the nucleation of giant vortices. We also studied the system of five electrons and show that the mechanism of the flip of the vortex orientation found for three-electron system is reproduced for higher number of electrons.

Figure 1

(Color online) Positions of vortices of the conditional wave function calculated for the two electrons fixed at points $(\pm l_0,0)$ as function of the screening length $\alpha$. Solid lines correspond to lower horizontal axis and show the positions of vortices on the $x$ axis $(y=0)$. Dashed lines are plotted with respect to the upper horizontal axis and show the $y$ coordinate of vortices localized on the $x=-l_0$ line. All the results were obtained in the lowest Landau level (LLL) with the exception of the blue curves plotted in (b) calculated beyond the LLL approximation with a fully convergent basis set. At the left side of (b) we show the electron-vortex orientation before $(\alpha >0.44l_0)$ and after the formation of the giant vortex $(\alpha <0.44l_0)$. The electron positions are marked with dots, and the vortices by crosses.

Figure 2

(Color online) Contour plots of the logarithm of the absolute value of the reduced wave function calculated for angular momentum $L=15$ and the screening length $\alpha =l_0∕2$ in the LLL approximation for the pinned electron positions $(\pm l_0,0)$ (a), $(\pm l_0∕10,0)$ (b), and $(\pm l_0+l_0∕2,0)$ (c). Electron positions are marked by blue arrows in (a).

Figure 3

(Color online) Vortices and the conditional probability for $L=9$ when the two electrons are pinned at $(-l_0,0)$ and $(-l_0∕2,0)$ (black solid lines). The two outermost vortices for electrons in pinned at $(\pm l_0,0)$ are shown with dashed curves. The reduced Laughlin wave functions along the $y=0$ axis are also shown by red and blue lines, for the symmetrically and nonsymmetrically pinned electrons, respectively.

Figure 4

Overlap integral of the conditional wave functions calculated for the lowest-energy state diagonalizing the Hamiltonian in the LLL subspace and the state described by the Laughlin wave function (5). In (a) two of the electrons are fixed at positions $(\pm l_0,0)$, and in (b) at $(-l_0,0)$ and $(-l_0∕2,0)$.

Figure 5

Contour plots of the absolute value of the reduced wave function for the state with $L=15$ and the two electrons fixed at $(-l_0,0)$ and $(-l_0∕2,0)$ (indicated by crosses). (a)–(c) show the ED wave function as obtained within the LLL approximation for different values of $\alpha$, (d) shows the Laughlin wave function corresponding to this state. One of the vortices bound to the test electron which crosses through the wave function’s maximum is indicated by a star in (a) and (c), while in (b) it creates a distinct minimum at $x=3l_0$.

Figure 6

Evolution of the absolute value of the reduced wave function at $y=0$ as function of $\alpha$ for the state with $L=15$ and two electrons pinned in $(-l_0,0)$ and $(-l_0∕2,0)$.

Figure 7

(Color online) Coefficient $\kappa$ determining the asymptotic dependence of the pair correlation function $W(z_a,z_b)\simeq {\mid z_a-z_b\mid }^{\kappa }$ for $z_a\to z_b$ (see the text) for different total angular momentum states. Solid curves were calculated as fits to the actual PCF values calculated on an arc of length $0.6l_0$ for $\mid z_a\mid =1.5l_0$, and the dashed curves on a line segment of length $0.2l_0$ with one of the ends fixed at the origin for $z_a=0$.

Figure 8

(Color online) Contour plots of the logarithm of the absolute value of the $L=12$ reduced wave function when the two electrons are pinned at $(\pm l_0,0)$ calculated with a basis of 5158 Slater determinants including higher Landau levels. Plots (a)–(h) correspond to the screening lengths $\alpha ∕l_0=0.421$, 0.417, 0.416, 0.415, and 0.412, respectively. Red arrows point to the antivortex positions.

Figure 9

(Color online) Zoom of Fig. 1b. Solid curves show the $x$-position coordinates of the vortices localized at the $y=0$ axis and the dashed lines are the $y$-position coordinates of vortices localized at $x=\pm l_0$ line. Black curves correspond to the LLL approximation and the exact results are plotted in blue.

Figure 10

(Color online) Contour plots of the logarithm of the absolute value of the reduced wave function calculated for five electrons at angular momentum $L=35$ in the LLL approximation. Plot (a) corresponds to the Coulomb interaction potential and plots (b)–(d) to the screening lengths $\alpha =0.0889l_0$, $\alpha =0.0643l_0$, and $\alpha =0.0222l_0$. Positions of four electrons are pinned at the corners of the square $(\pm l_0,\pm l_0)$. Electron positions are marked by blue arrows.